Geometry optimization
Fixed cell
We use DFTK and the GeometryOptimization package to find the minimal-energy bond length of the $H_2$ molecule. First we set up an appropriate DFTKCalculator (see AtomsCalculators integration), for which we use the LDA model just like in the Tutorial for silicon in combination with a pseudodojo pseudopotential (see Pseudopotentials).
using DFTK
using PseudoPotentialData
pseudopotentials = PseudoFamily("dojo.nc.sr.pbe.v0_4_1.standard.upf")
calc = DFTKCalculator(;
model_kwargs = (; functionals=LDA(), pseudopotentials), # model_DFT keyword arguments
basis_kwargs = (; kgrid=[1, 1, 1], Ecut=10) # PlaneWaveBasis keyword arguments
)DFTKCalculator(functionals=Xc(lda_x, lda_c_pw), pseudopotentials=PseudoFamily("dojo.nc.sr.pbe.v0_4_1.standard.upf"), Ecut=10, kgrid=[1, 1, 1])Next we set up an initial hydrogen molecule within a box of vacuum. We use the parameters of the equivalent tutorial from ABINIT and DFTK's AtomsBase integration to setup the hydrogen molecule. We employ a simulation box of 10 bohr times 10 bohr times 10 bohr and a pseudodojo pseudopotential.
using LinearAlgebra
using Unitful
using UnitfulAtomic
r0 = 1.4 # Initial bond length in Bohr
a = 10.0 # Box size in Bohr
cell_vectors = [[a, 0, 0]u"bohr", [0, a, 0]u"bohr", [0, 0, a]u"bohr"]
h2_crude = periodic_system([:H => [0, 0, 0.]u"bohr",
:H => [r0, 0, 0]u"bohr"],
cell_vectors)FlexibleSystem(H₂, periodicity = TTT):
cell_vectors : [ 10 0 0;
0 10 0;
0 0 10]u"a₀"
Atom(H, [ 0, 0, 0]u"a₀")
Atom(H, [ 1.4, 0, 0]u"a₀")
Finally we call minimize_energy! to start the geometry optimisation. We use verbosity=2 to get some insight into the minimisation. With verbosity=1 only a summarising table would be printed and with verbosity=0 (default) the minimisation would be quiet.
using GeometryOptimization
results = minimize_energy!(h2_crude, calc; tol_forces=2e-6, verbosity=2)
nothing # hiden Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -1.110722597715 -0.82 0.80 7.0 5.24s
2 -1.117204752835 -2.19 -1.88 0.80 1.0 2.25s
3 -1.117250547216 -4.34 -2.71 0.80 1.0 18.9ms
4 -1.117251167373 -6.21 -3.54 0.80 1.0 18.9ms
5 -1.117251188561 -7.67 -4.09 0.80 1.0 56.4ms
6 -1.117251189922 -8.87 -4.50 0.80 1.0 19.5ms
7 -1.117251190134 -9.67 -5.38 0.80 1.0 19.8ms
8 -1.117251190143 -11.05 -6.29 0.80 1.0 20.4ms
9 -1.117251190143 -12.72 -6.73 0.80 1.0 29.7ms
10 -1.117251190143 -14.18 -7.75 0.80 1.0 20.5ms
11 -1.117251190143 -15.65 -7.72 0.80 2.0 24.9ms
12 -1.117251190143 -15.05 -9.32 0.80 1.0 20.7ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -1.117251190143 -9.06 0.80 1.0 4.02s
2 -1.117251190143 -15.35 -9.97 0.80 1.0 18.4ms
Geometry optimisation convergence (in atomic units)
┌─────┬─────────────────┬───────────┬─────────────┬────────┐
│ n │ Energy │ log10(ΔE) │ max(Force) │ Δtime │
├─────┼─────────────────┼───────────┼─────────────┼────────┤
│ 0 │ -1.117251190143 │ │ 0.0269318 │ 22.9s │
└─────┴─────────────────┴───────────┴─────────────┴────────┘
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -1.117517998097 -2.46 0.80 1.0 13.1ms
2 -1.117520586052 -5.59 -3.49 0.80 1.0 18.3ms
3 -1.117520612985 -7.57 -4.23 0.80 1.0 18.4ms
4 -1.117520613449 -9.33 -5.14 0.80 1.0 18.7ms
5 -1.117520613464 -10.81 -5.92 0.80 1.0 19.0ms
6 -1.117520613465 -12.13 -6.42 0.80 1.0 19.3ms
7 -1.117520613465 -13.35 -7.39 0.80 1.0 57.1ms
8 -1.117520613465 -15.35 -8.19 0.80 1.0 20.8ms
9 -1.117520613465 -15.35 -8.98 0.80 1.0 21.0ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -1.118247264594 -1.69 0.80 1.0 13.0ms
2 -1.118339043935 -4.04 -2.70 0.80 1.0 18.5ms
3 -1.118340011114 -6.01 -3.44 0.80 1.0 30.5ms
4 -1.118340028555 -7.76 -4.36 0.80 1.0 18.7ms
5 -1.118340029140 -9.23 -5.12 0.80 1.0 19.1ms
6 -1.118340029166 -10.57 -5.62 0.80 1.0 19.3ms
7 -1.118340029168 -11.73 -6.68 0.80 1.0 19.9ms
8 -1.118340029168 -13.52 -7.58 0.80 1.0 25.7ms
9 -1.118340029168 -15.05 -8.12 0.80 1.0 20.3ms
10 -1.118340029168 + -Inf -8.55 0.80 1.0 20.4ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -1.118340029168 -9.05 0.80 1.0 13.0ms
2 -1.118340029168 + -Inf -10.36 0.80 1.0 18.2ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -1.118340029168 -11.04 0.80 1.0 12.9ms
2 -1.118340029168 + -15.65 -11.41 0.80 1.0 24.4ms
┌─────┬─────────────────┬───────────┬─────────────┬────────┐
│ n │ Energy │ log10(ΔE) │ max(Force) │ Δtime │
├─────┼─────────────────┼───────────┼─────────────┼────────┤
│ 1 │ -1.118340029168 │ │ 0.00297055 │ 2.89s │
└─────┴─────────────────┴───────────┴─────────────┴────────┘
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -1.118346721508 -3.09 0.80 1.0 13.0ms
2 -1.118346864760 -6.84 -4.11 0.80 1.0 25.5ms
3 -1.118346866237 -8.83 -4.86 0.80 1.0 18.3ms
4 -1.118346866262 -10.59 -5.78 0.80 1.0 24.4ms
5 -1.118346866263 -12.08 -6.55 0.80 1.0 19.4ms
6 -1.118346866263 -13.42 -7.04 0.80 1.0 19.4ms
7 -1.118346866263 -14.88 -8.13 0.80 1.0 23.0ms
8 -1.118346866263 + -Inf -8.95 0.80 1.0 19.9ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -1.118354724004 -2.61 0.80 1.0 12.8ms
2 -1.118356087056 -5.87 -3.62 0.80 1.0 18.1ms
3 -1.118356101131 -7.85 -4.37 0.80 1.0 21.3ms
4 -1.118356101377 -9.61 -5.29 0.80 1.0 18.7ms
5 -1.118356101385 -11.10 -6.06 0.80 1.0 21.6ms
6 -1.118356101386 -12.45 -6.55 0.80 1.0 21.1ms
7 -1.118356101386 -13.60 -7.65 0.80 1.0 22.4ms
8 -1.118356101386 + -14.65 -8.45 0.80 1.0 20.2ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -1.118356101386 -9.24 0.80 1.0 15.7ms
2 -1.118356101386 -14.37 -9.82 0.80 1.0 18.5ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -1.118356101386 -10.62 0.80 1.0 13.1ms
2 -1.118356101386 -14.15 -11.46 0.80 1.0 18.7ms
┌─────┬─────────────────┬───────────┬─────────────┬────────┐
│ n │ Energy │ log10(ΔE) │ max(Force) │ Δtime │
├─────┼─────────────────┼───────────┼─────────────┼────────┤
│ 2 │ -1.118356101386 │ -4.79 │ 4.47268e-5 │ 532ms │
└─────┴─────────────────┴───────────┴─────────────┴────────┘
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -1.118356102848 -4.94 0.80 1.0 13.3ms
2 -1.118356102877 -10.54 -5.96 0.80 1.0 18.5ms
3 -1.118356102877 -12.52 -6.70 0.80 1.0 25.0ms
4 -1.118356102877 -14.22 -7.62 0.80 1.0 19.1ms
5 -1.118356102877 + -14.40 -8.39 0.80 1.0 25.0ms
6 -1.118356102877 -14.42 -8.89 0.80 1.0 19.7ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -1.118356104771 -4.40 0.80 1.0 15.9ms
2 -1.118356105116 -9.46 -5.42 0.80 1.0 18.4ms
3 -1.118356105120 -11.45 -6.17 0.80 1.0 18.5ms
4 -1.118356105120 -13.20 -7.09 0.80 1.0 21.6ms
5 -1.118356105120 -15.35 -7.86 0.80 1.0 19.3ms
6 -1.118356105120 + -14.65 -8.35 0.80 1.0 22.2ms
7 -1.118356105120 -14.15 -9.45 0.80 1.0 20.5ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -1.118356105120 -10.08 0.80 1.0 13.0ms
2 -1.118356105120 + -15.35 -11.02 0.80 1.0 18.4ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -1.118356105120 -11.67 0.80 1.0 12.9ms
2 -1.118356105120 -14.35 -12.44 0.80 1.0 18.4ms
┌─────┬─────────────────┬───────────┬─────────────┬────────┐
│ n │ Energy │ log10(ΔE) │ max(Force) │ Δtime │
├─────┼─────────────────┼───────────┼─────────────┼────────┤
│ 3 │ -1.118356105120 │ -8.43 │ 9.90989e-9 │ 474ms │
└─────┴─────────────────┴───────────┴─────────────┴────────┘
Structure after optimisation (note that the atom has wrapped around)
results.systemFlexibleSystem(H₂, periodicity = TTT):
cell_vectors : [ 10 0 0;
0 10 0;
0 0 10]u"a₀"
Atom(H, [-0.0431829, -6.81322e-18, -1.43583e-17]u"a₀")
Atom(H, [ 1.44318, -1.26182e-17, -1.30587e-17]u"a₀")
Compute final bond length:
rmin = norm(position(results.system[1]) - position(results.system[2]))
println("Optimal bond length: ", rmin)Optimal bond length: 1.4863658688992982 a₀
Our result (1.486 Bohr) agrees with the equivalent tutorial from ABINIT.
Variable cell
Recent versions of GeometryOptimization support cell optimization as well by passing variablecell=true to minimize_energy!.
For a plane-wave code like DFTK, variable cells pose an additional challenge: changes to the cell's size affect the used plane waves, leading to discontinuities in the energy and stresses. This can cause the geometry optimization to struggle and/or fail to converge.
A practical strategy to overcome this problem is Energy cutoff smearing. As a demonstration let us find the optimal lattice constant of silicon.
Like before we define a calculator, this time with a kinetic_blowup set to use energy cutoff smearing:
calc = DFTKCalculator(;
model_kwargs = (; functionals=LDA(), pseudopotentials,
kinetic_blowup=BlowupCHV()),
basis_kwargs = (; kgrid=[2, 2, 2], Ecut=10)
)DFTKCalculator(functionals=Xc(lda_x, lda_c_pw), pseudopotentials=PseudoFamily("dojo.nc.sr.pbe.v0_4_1.standard.upf"), Ecut=10, kgrid=[2, 2, 2])And here is our starting silicon structure:
a = 10.0u"bohr" # Approximate Silicon lattice constant
cell_vectors = a/2 * [[0, 1, 1], [1, 0, 1], [1, 1, 0]]
initial_silicon = periodic_system([:Si => ones(3)/8,
:Si => -ones(3)/8],
cell_vectors;
fractional=true)FlexibleSystem(Si₂, periodicity = TTT):
cell_vectors : [ 0 5 5;
5 0 5;
5 5 0]u"a₀"
Atom(Si, [ 1.25, 1.25, 1.25]u"a₀")
Atom(Si, [ -1.25, -1.25, -1.25]u"a₀")
We now minimize, passing variablecell=true:
using GeometryOptimization
results = minimize_energy!(initial_silicon, calc; variablecell=true,
tol_virial=2e-6, verbosity=2)
nothing # hiden Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -8.335328316711 -0.81 0.80 8.0 731ms
2 -8.340897479572 -2.25 -1.69 0.80 1.3 621ms
3 -8.341205177135 -3.51 -2.76 0.80 2.3 32.2ms
4 -8.341214238069 -5.04 -3.31 0.80 3.7 40.7ms
5 -8.341214269308 -7.51 -4.08 0.80 1.7 28.2ms
6 -8.341214272498 -8.50 -5.60 0.80 2.3 33.1ms
7 -8.341214272517 -10.71 -5.84 0.80 4.7 45.2ms
8 -8.341214272518 -12.23 -6.78 0.80 1.3 43.8ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -8.341214272518 -7.63 0.80 1.0 591ms
2 -8.341214272518 -14.05 -8.05 0.80 1.0 25.1ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -8.341214272518 -8.87 0.80 1.0 22.8ms
2 -8.341214272518 + -14.75 -8.82 0.80 1.0 25.1ms
Geometry optimisation convergence (in atomic units)
┌─────┬─────────────────┬───────────┬─────────────┬─────────────┬──────────┬────
│ n │ Energy │ log10(ΔE) │ max(Force) │ max(Virial) │ Pressure │ ⋯
├─────┼─────────────────┼───────────┼─────────────┼─────────────┼──────────┼────
│ 0 │ -8.341214272518 │ │ 3.53734e-15 │ 0.163458 │ -0.16 │ ⋯
└─────┴─────────────────┴───────────┴─────────────┴─────────────┴──────────┴────
1 column omitted
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -8.350575043294 -1.21 0.80 8.7 107ms
2 -8.351863854058 -2.89 -1.88 0.80 1.3 28.4ms
3 -8.351926770042 -4.20 -2.79 0.80 3.3 38.8ms
4 -8.351928505709 -5.76 -3.61 0.80 3.3 39.8ms
5 -8.351928514075 -8.08 -4.55 0.80 2.3 34.2ms
6 -8.351928514627 -9.26 -5.27 0.80 4.0 60.1ms
7 -8.351928514646 -10.73 -6.26 0.80 3.3 40.0ms
8 -8.351928514646 -12.52 -7.47 0.80 4.0 43.7ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -8.353396040153 -1.52 0.80 8.0 64.9ms
2 -8.353731656749 -3.47 -2.21 0.80 1.0 30.9ms
3 -8.353757401899 -4.59 -2.98 0.80 3.7 41.6ms
4 -8.353757523421 -6.92 -4.24 0.80 2.3 32.8ms
5 -8.353757531682 -8.08 -4.94 0.80 3.3 41.2ms
6 -8.353757531724 -10.37 -5.76 0.80 3.0 39.5ms
7 -8.353757531726 -11.78 -7.01 0.80 2.3 34.7ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -8.353757531726 -8.16 0.80 1.0 23.5ms
2 -8.353757531726 -14.45 -8.63 0.80 1.0 30.2ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -8.353757531726 -8.96 0.80 1.0 22.8ms
2 -8.353757531726 + -Inf -9.49 0.80 1.0 25.9ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -8.353757531726 -9.54 0.80 1.0 23.2ms
2 -8.353757531726 -14.45 -10.05 0.80 1.0 26.1ms
┌─────┬─────────────────┬───────────┬─────────────┬─────────────┬──────────┬────
│ n │ Energy │ log10(ΔE) │ max(Force) │ max(Virial) │ Pressure │ ⋯
├─────┼─────────────────┼───────────┼─────────────┼─────────────┼──────────┼────
│ 1 │ -8.353757531726 │ │ 9.87418e-15 │ 0.0156413 │ -0.016 │ ⋯
└─────┴─────────────────┴───────────┴─────────────┴─────────────┴──────────┴────
1 column omitted
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -8.353723816134 -3.20 0.80 7.7 64.4ms
2 -8.353760134973 -4.44 -3.87 0.80 1.0 25.7ms
3 -8.353770535634 -4.98 -3.93 0.80 5.7 59.1ms
4 -8.353770537265 -8.79 -4.30 0.80 1.0 25.8ms
5 -8.353770537313 -10.32 -4.72 0.80 2.0 30.9ms
6 -8.353770537401 -10.06 -5.70 0.80 1.7 29.1ms
7 -8.353770537403 -11.69 -6.79 0.80 3.3 42.3ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -8.353743564644 -2.00 0.80 9.0 67.6ms
2 -8.353866136658 -3.91 -2.71 0.80 1.0 25.6ms
3 -8.353895037595 -4.54 -3.27 0.80 4.7 51.3ms
4 -8.353895102223 -7.19 -4.02 0.80 2.0 31.9ms
5 -8.353895110584 -8.08 -4.88 0.80 2.3 34.5ms
6 -8.353895110769 -9.73 -5.77 0.80 3.3 39.6ms
7 -8.353895110771 -11.71 -6.47 0.80 3.3 43.4ms
8 -8.353895110771 -13.30 -7.48 0.80 2.3 33.5ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -8.353895110771 -8.09 0.80 1.0 23.3ms
2 -8.353895110771 + -Inf -8.94 0.80 1.0 30.7ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -8.353895110771 -9.37 0.80 1.0 23.0ms
2 -8.353895110771 -14.45 -9.58 0.80 1.0 26.0ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -8.353895110771 -9.95 0.80 1.0 23.3ms
2 -8.353895110771 + -Inf -10.08 0.80 1.0 25.8ms
┌─────┬─────────────────┬───────────┬─────────────┬─────────────┬──────────┬────
│ n │ Energy │ log10(ΔE) │ max(Force) │ max(Virial) │ Pressure │ ⋯
├─────┼─────────────────┼───────────┼─────────────┼─────────────┼──────────┼────
│ 2 │ -8.353895110771 │ -3.86 │ 7.64136e-15 │ 0.000646179 │ -0.00065 │ ⋯
└─────┴─────────────────┴───────────┴─────────────┴─────────────┴──────────┴────
1 column omitted
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -8.353788719641 -3.55 0.80 8.7 67.0ms
2 -8.353871278963 -4.08 -3.77 0.80 1.0 25.8ms
3 -8.353895211589 -4.62 -3.91 0.80 6.0 55.2ms
4 -8.353895212347 -9.12 -4.11 0.80 1.0 26.0ms
5 -8.353895212406 -10.23 -4.61 0.80 1.3 31.1ms
6 -8.353895212476 -10.16 -4.82 0.80 1.0 26.5ms
7 -8.353895212512 -10.43 -5.44 0.80 1.3 27.8ms
8 -8.353895212517 -11.36 -6.05 0.80 3.3 39.3ms
9 -8.353895212516 + -12.71 -5.73 0.80 3.0 42.0ms
10 -8.353895212517 -12.37 -6.53 0.80 1.0 27.1ms
11 -8.353895212517 -13.25 -7.78 0.80 2.0 33.1ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -8.353799009186 -3.29 0.80 8.7 70.1ms
2 -8.353874257644 -4.12 -3.78 0.80 1.0 25.7ms
3 -8.353895362124 -4.68 -3.86 0.80 6.0 55.6ms
4 -8.353895362560 -9.36 -4.22 0.80 1.0 25.8ms
5 -8.353895362648 -10.06 -4.56 0.80 1.3 32.2ms
6 -8.353895362676 -10.55 -4.94 0.80 1.0 26.3ms
7 -8.353895362685 -11.04 -5.52 0.80 1.7 29.5ms
8 -8.353895362686 -12.05 -6.88 0.80 2.0 32.1ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -8.353895362686 -7.58 0.80 1.0 23.5ms
2 -8.353895362686 -13.97 -7.99 0.80 1.0 26.1ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -8.353895362686 -8.45 0.80 1.0 23.0ms
2 -8.353895362686 + -Inf -8.61 0.80 1.0 31.2ms
n Energy log10(ΔE) log10(Δρ) α Diag Δtime
--- --------------- --------- --------- ---- ---- ------
1 -8.353895362686 -9.01 0.80 1.0 23.0ms
2 -8.353895362686 -14.45 -9.09 0.80 1.0 26.5ms
┌─────┬─────────────────┬───────────┬─────────────┬─────────────┬──────────┬────
│ n │ Energy │ log10(ΔE) │ max(Force) │ max(Virial) │ Pressure │ ⋯
├─────┼─────────────────┼───────────┼─────────────┼─────────────┼──────────┼────
│ 3 │ -8.353895362686 │ -6.60 │ 5.12015e-15 │ 1.71583e-6 │ -1.7e-6 │ ⋯
└─────┴─────────────────┴───────────┴─────────────┴─────────────┴──────────┴────
1 column omitted
Structure after optimization
results.systemFlexibleSystem(Si₂, periodicity = TTT):
cell_vectors : [-2.76298e-18 5.27783 5.27783;
5.27783 -2.76298e-18 5.27783;
5.27783 5.27783 -2.76298e-18]u"a₀"
Atom(Si, [ 1.31946, 1.31946, 1.31946]u"a₀")
Atom(Si, [-1.31946, -1.31946, -1.31946]u"a₀")
Since here the cell was rescaled but its shape did not change, we directly extract the optimal lattice constant from one of the cell vectors:
using AtomsBase
amin = AtomsBase.cell_vectors(results.system)[1][2]*2
println("Optimal lattice constant: ", amin)Optimal lattice constant: 10.555662923271829 a₀
Note that while for silicon the positions of the atoms are fixed by symmetry, in general a variable cell optimization will try to optimize both the cell and the positions of the individual atoms.